## 2. Free-free emission from a windFor a spherically symmetric wind the equation for mass conservation reads where is the mass loss rate,
is the mass density, while the ion density is given by The average number of electrons per ion is given by and the mean atomic mass per ion is given by The fraction of neutral hydrogen is given by . Since there is no information available from observations
concerning the ionization balance in the wind, the most plausible
assumption one can make is that, after an initial acceleration of the
wind, the ionization balance is frozen-in due to the rapid decrease of
the density. The ionization balance in the wind then reflects the
ionization balance at the base of the wind which is determined by
collisional ionization equilibrium. We determined the values of
, ,
,
The linear free-free absorption coefficient is given by (Allen, 1973) The second part of this expression can be written as with For calculating the gaunt factors we followed the procedure outlined by Waters and Lamers (1984) with the exception that Lamers and Waters write with the mean value of the squared atomic charge. In our case separating out would be impractical given the form of Eq. (6). For low values of we use the expression for the gaunt factor given by Allen (1973) but with the correction discussed by Leitherer and Robert (1991). For high values of we use the expression by Mewe et al. (1986) (see also Gronenschild and Mewe, 1978). In practice we took the maximum value for , at a given temperature and frequency, which resulted from these approximations. The results are shown in Fig. 2.
By introducing the dimensionless parameters and , the absorption coefficient can be written as The constant is independent of the frequency. The total flux emitted by the star and the wind is easily calculated following the standard procedure outlined by Wright and Barlow (1975), Panagia and Felli (1975) and Lamers and Waters (1984). The total flux is given by where , ,
is the Planck function,
is the black-body temperature of the star, is
the wind temperature and The optical depth depends of course on the assumed velocity law of the wind for which no information is available from observations. Therefore we use in this paper a velocity law for a wind which is accelerating up to some distance and then obtains its final velocity The acceleration of the wind is determined by the value of . The dimensionless velocity is given by for and for . Given a velocity law for the wind, the run of the ion density in
the wind can be determined from Eq. (1).
In Table 2 the related expressions for are
listed. Also listed in Table 2 are the expressions for the
emission measure
With the expressions for , as given in Table 2, part of Eq. (10) can be evaluated analytically resulting in with and the incomplete Gamma function (Abramowitz and Stegun, 1968). The first term on the right accounts for the emission by the star, which can be attenuated by absorption due to the wind. The second term accounts for the emission from the cone in front of the star while the third term accounts for the emission from the wind acceleration region outside the previously mentioned cone. The fourth term described the emission from the volume (outside the cone) where the wind has reached its terminal velocity. Note that for a wind with constant velocity () the third term does not contribute. At low frequencies, at which is large, only the last term effectively contributes to the flux resulting in which is identical to the expression found by Wright and Barlow (1975). At higher frequencies, at which becomes small, we can make the approximation in Eqs. (10) and (11). Evaluation of the resulting integrals shows that, at frequencies at which the emission is optically thin, the flux is given by with given in Table 2 and the free-free emissivity (e.g. Rybicki and Lightman, 1979) The last identity in Eq. (13) results from applying
Kirchhoff's law. As could be anticipated, in the optically thin part
of the spectrum the flux is composed of the contribution by the star
and the free-free emission from a wind with an emission measure
For completeness we present in Table 2 also the effective radius and the effective optical depth (see Wright and Barlow, 1975). The effective radius is defined by assuming that the emission at a given frequency originates from the volume at with given by Eq. (10). The effective optical depth is then defined by . © European Southern Observatory (ESO) 1997 Online publication: July 3, 1998 |